The Wigner distribution function applied to optical signals and systems

In this paper the Wigner distribution function has been introduced for optical signals and systems. The Wigner distribution function of an optical signal appears to be in close resemblance to the ray concept in geometrical optics. This resemblance reaches even farther: although derived from Fourier optics, the Wigner distribution functions of some elementary optical systems can directly be interpreted in terms of geometrical optics.     

The Wigner distribution function

In this letter, the relationship between the characteristic function for two arbitrary noncommuting observables and a generalized Wigner distribution function is established. This distribution function is shown to have no simple interpretation in the sense of probability theory but, in lieu of its special properties, can be used directly for calculating the expectation values of observables.     

Heterodyne measurement of Wigner distributions for classical optical fields

We demonstrate a two-window heterodyne method for measuring the x-p cross correlation, ??(*)(x)? (p)?, of an optical field ? for transverse position x and transverse momentum p. This scheme permits independent control of the x and p resolution. A simple linear transform of the x-p correlation function yields the Wigner phase-space distribution. This technique is useful for both coherent and low-coherence light sources and may permit new biological imaging techniques based on transverse coherence measurement with time gating. We point out an interesting analogy between x-p correlation measurements for classical-wave and quantum fields.     

Wigner分布的模糊处理

针对Wigner分布在应用中物理意义含混不清等特点,提出时频分布具有模糊性的观点。将模糊集的概念和方法引入对时频分析结果的理解和再处理,提出了模糊时频点的概念和对Wigner分布进行模糊点处理的方法,将此方法用于实际信号的处理,效果良好。     

The Wigner distribution function applied to laser radiation

The Wigner distribution function (WDF) is investigated analytically and then calculated numerically and represented graphically for Laguerre-Gaussian modes and for a laser with annular gain medium. The derivation of the normalized beam qualityM ² from the WDF on the basis of second order moments is discussed, andM ² is determined for the examples mentioned above.     

The Wigner distribution function—50th birthday

Foundations of Physics - We discuss the profound influence which the Wigner distribution function has had in many areas of physics during its fifty years of existence.     

On the Wigner distribution function for an oscillator

We present two new derivations of the Wigner distribution function for a simple harmonic oscillator Hamiltonian. Both methods are facilitated using a formula which expresses the Wigner function as a simple trace. The first method of derivation then utilizes a modification of a theorem due to Messiah. An alternative procedure makes use of the coherent state representation of an oscillator. The Wigner distribution function gives a semiclassical joint probability for finding the system with given coordinates and momenta, and the joint probability is factorable for the special case of an oscillator. An important application of this result occurs in the theory of nuclear fission for calculating the probability distributions for the masses, kinetic energies, and vibrational energies of the fission fragments at infinite separation.     

激光光路追踪Wigner分布函数方法

提出了一种考虑衍射效应的激光几何光路追踪方法。引入由激光标量场定义的Wigner激光能量相空间分布函数,并给出该函数满足的刘维尔运动方程。Wigner分布函数用来描述经过空间任一点沿任一方向传输的激光光线上的能量分配。激光能量沿由波包色散关系定义的光线轨迹保持不变（真空中）或者衰减（等离子体中）。与传统几何光路追踪方法相比,该方法从理论上给出了激光光线初始携带能量份额的计算方法,并且将激光标量场的相位信息自然地包含在Wigner分布函数的定义里。算例表明,该方法与解析模型及广泛使用的菲涅耳衍射积分方法结果一致。     

激光光路追踪Wigner分布函数方法

提出了一种考虑衍射效应的激光几何光路追踪方法。引入由激光标量场定义的Wigner激光能量相空间分布函数,并给出该函数满足的刘维尔运动方程。Wigner分布函数用来描述经过空间任一点沿任一方向传输的激光光线上的能量分配。激光能量沿由波包色散关系定义的光线轨迹保持不变（真空中）或者衰减（等离子体中）。与传统几何光路追踪方法相比,该方法从理论上给出了激光光线初始携带能量份额的计算方法,并且将激光标量场的相位信息自然地包含在Wigner分布函数的定义里。算例表明,该方法与解析模型及广泛使用的菲涅耳衍射积分方法结果一致。     

The van der Waals limit for classical systems

For a -dimensional system of particles with the two-body potentialq(r)+ ^v K(r) and density , it is proved under fairly weak conditions onq andK that the canonical pressure (, ) and chemical potential (, ) tend to definite limits when 0. The limiting functions are absolutely continuous and are given in terms of the derivative of the limiting free energy density which was found in Part I.     

Joint Wigner distribution for spin-1/2 particles

The Wigner distribution has proven to be a useful tool in many quantum problems requiring a joint distribution of position and momentum. In the present paper we develop a joint distribution for spin components within the spirit of the Wigner distribution. This distribution provides an insight into the quantum theory of measurement. We also discuss how one may write joint distributions for two arbitrary noncommuting operators.It is a pleasure to dedicate this paper to Professor John A. Wheeler—physicist, philosopher, poet—an inspiration to us all.Work supported by the Office of Naval Research.     

Equilibrium states for classical systems

It is shown that the Dobrushin-Lanford-Ruelle equations for the probability measure μ, and the Kirkwood-Salsburg type equations for the lattice or continuum correlation functions ?, and for the spin correlation functions σ, are essentially equivalent for all ?, σ, and μ satisfying certain boundedness conditions. It is also noted that the lattice equations are identical to the equations for the stationary states of a certain Markoff process. This extends previous results of Ruelle, Brascamp and Holley who proved some of these equivalences for states.     

非对易相空间中阻尼系统的Wigner函数

用量子力学来处理经典的阻尼系统,考虑到空间变量对易关系中包含的坐标-坐标和动量-动量的非对易性,利用Wigner函数在非对易相空间的基本性质,得到了阻尼谐振子在非对易相空间中的Wigner函数与对易空间及非对易空间的形式一致.     

The equivalence of ensembles for classical systems of particles

For systems of particles in classical phase space with standard Hamiltonian, we consider (spatially averaged) microcanonical Gibbs distributions in finite boxes. We show that infinite-volume limits along suitable subsequences exist and are grand canonical Gibbs measures. On the way, we establish a variational formula for the thermodynamic entropy density, as well as a variational characterization of grand canonical Gibbs measures.